Question

Graph each function. Based on the graph, state the domain and the range, and find any intercepts. $$ f(x)=\left\\{\begin{array}{ll} -\ln x & \text { if } 0

Short answer

The domain is \(0 < x < \infty\), the range is \(-\infty < f(x) < \infty\), and there are no intercepts.

Step-by-step solution

Understand the Piecewise Function

The function is defined in two parts. For values of x between 0 and 1, the function is given by \(-\ln(x)\). For values of x greater than or equal to 1, the function is \(\ln(x)\). These definitions will help determine the graph's shape.

Graph Each Piece of the Function Separately

First, graph \(-\ln(x)\) for \(0 < x < 1\). Next, graph \(\ln(x)\) for \(x \geq 1\).

Combine the Graphs

Plot the two segments on the same set of axes. Ensure continuity by observing the behavior at the boundaries (x = 1).

State the Domain

The domain is the set of x-values over which the function is defined. Here, it is \(0 < x < \infty\).

State the Range

The range is the set of possible output values. From the graph, it can be observed that the range of the function is all real numbers \(-\infty < f(x) < \infty\).

Find Any Intercepts

Check where the graph crosses the axes to determine any x or y intercepts. From the combined graph, there are no intercepts.

Key concepts

graphing functions

Graphing functions involves plotting the output of a function for various input values on a set of axes. For piecewise functions, each part of the function is graphed according to its specific rule over its designated interval.
Let's break down the approach for the given piecewise function:

• For the interval where \(0 < x < 1\), the function is \(-\ln(x)\). This represents a downward curve since the natural logarithm of a number between 0 and 1 is negative, and taking the negative of a negative number results in a positive value.
• For the interval where \(x \geq 1\), the function is \(\ln(x)\). This represents the standard logarithmic curve, which increases slowly and remains positive for all \(x\) values in its domain.

To combine these graphs, plot \(\text{-ln(x)}\) for \(0 < x < 1\) and \(\text{ln(x)}\) for \(x \geq 1\) on the same set of axes. Ensure that the graph is continuous at \(x = 1\), which is the boundary point between the intervals.

domain and range

The domain and range of a function describe the set of possible input and output values, respectively.
For this function, the domain is all \(x\)-values where the function is defined.

• For \(\text{-ln(x)}\), \(x\) must be between 0 and 1 (excluding 0).
• For \(\text{ln(x)}\), \(x\) must be greater than or equal to 1.

Therefore, the domain of the entire piecewise function is \(0 < x < \infty\).
The range is the set of possible output values, \(f(x)\):

• For \(0 < x < 1\), \(\text{-ln(x)}\) ranges from 0 (approaching \(x = 1\)) to \(\infty\) (approaching \(x = 0\)).
• For \(x \geq 1\), \(\text{ln(x)}\) ranges from 0 (at \(x = 1\)) to \(\infty\) (as \(x\) increases).

Combining these, the range of the piecewise function is all real numbers \(-\infty < f(x) < \infty\).

natural logarithm

The natural logarithm is a special logarithm with the base \(e\) (where \(e \approx 2.71828\)). It's denoted as \(\text{ln(x)}\).
Key properties of the natural logarithm include:

• \(\text{ln(1) = 0}\)
• \(\text{ln(e) = 1}\)
• For \(0 < x < 1\), \(\ln(x)\) is negative, decreasing, and approaches \(-\infty\) as \(x\) approaches 0.
• For \(x > 1\), \(\text{ln(x)}\) is positive and increases slowly without bound as \(x\) increases.

In the given piecewise function:

• \(-\ln(x)\) for \( 0 < x < 1\) flips the natural logarithm curve upside down, making it positive.
• \(\ln(x)\) for \(x \geq 1\) behaves as usual, starting from \(0\) at \(x = 1\) and increasing slowly.

Understanding how these parts combine helps in interpreting the behavior of the whole piecewise function.